Integrand size = 17, antiderivative size = 114 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}-\frac {3 d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {44, 65, 214} \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=-\frac {3 d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}}+\frac {3 d \sqrt {c+d x}}{4 (a+b x) (b c-a d)^2}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)} \]
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Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}-\frac {(3 d) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{4 (b c-a d)} \\ & = -\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac {(3 d) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 (b c-a d)^2} \\ & = -\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}-\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {1}{4} \left (\frac {\sqrt {c+d x} (-2 b c+5 a d+3 b d x)}{(b c-a d)^2 (a+b x)^2}+\frac {3 d^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{5/2}}\right ) \]
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Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) d^{2} \left (b x +a \right )^{2}}{4}+\frac {5 \sqrt {d x +c}\, \left (\frac {\left (3 d x -2 c \right ) b}{5}+a d \right ) \sqrt {\left (a d -b c \right ) b}}{4}}{\left (a d -b c \right )^{2} \left (b x +a \right )^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(102\) |
derivativedivides | \(2 d^{2} \left (\frac {\sqrt {d x +c}}{4 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{a d -b c}\right )\) | \(138\) |
default | \(2 d^{2} \left (\frac {\sqrt {d x +c}}{4 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\frac {3 \sqrt {d x +c}}{8 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{a d -b c}\right )\) | \(138\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (94) = 188\).
Time = 0.24 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.82 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {3 \, d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} - 5 \, \sqrt {d x + c} b c d^{2} + 5 \, \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
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Time = 0.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {\frac {5\,d^2\,\sqrt {c+d\,x}}{4\,\left (a\,d-b\,c\right )}+\frac {3\,b\,d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,{\left (a\,d-b\,c\right )}^2}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d}+\frac {3\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}} \]
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